Exponential inequalities

Last time

Before we start: last time, in Laws of logarithms and solving equations, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

The unknown is in the power

An inequality like three to the x is bigger than twenty cannot be solved by trial and error alone. The unknown sits in the power, so you need logarithms to bring it down. Take logs of both sides, collect the x terms, and divide. One careful check about signs decides whether the inequality flips.

Bring the power down

Take logarithms of both sides. Natural logs are usually easiest. The rule is that the log of a to the x equals x times the log of a. So three to the x greater than 20 becomes x ln 3 greater than ln 20, and x is greater than ln 20 over ln 3, about 2.73. Both sides must be positive before you take logs. A power of a positive number always is.

Does the inequality flip?

Taking logs keeps the inequality the same way, because the log function is increasing. But dividing by a negative number reverses it. The log of a number between 0 and 1 is negative. So take one half to the x less than 0.1. Then x ln 0.5 is less than ln 0.1. Divide by negative ln 0.5, and the sign flips: x is greater than 3.32. Check by trial: 0.5 to the 4 is 0.0625, which is below 0.1. It works.